Question 111
Let $k$ be an integer greater than or equal to $2$, and let $n_1$, $n_2$, $n_3$ be positive integers, and $a_1$, $a_2$, $a_3$ be integers greater than or equal to $1$ and less than or equal to $k-1$. Define
$$
b_i = a_i \sum_{j=0}^{n_i} k^j \quad (i = 1, 2, 3).
$$
Determine all possible combinations $(n_1, n_2, n_3)$ if $b_1 b_2 = b_3$.
$$
b_i = a_i \sum_{j=0}^{n_i} k^j \quad (i = 1, 2, 3).
$$
Determine all possible combinations $(n_1, n_2, n_3)$ if $b_1 b_2 = b_3$.